Definition 9.4.6.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. We say that $U$ is flat if, for every $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, the inclusion map
\[ \Lambda ^{2}_{1} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^2 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
is a categorical equivalence of simplicial sets.